Geophys. J. Int. (2008) doi: 10.1111/j.1365-246X.2008.03731.x GJI Seismology Rayleigh–Taylor instabilities with anisotropic lithospheric viscosity Einat Lev and Bradford H. Hager Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, Cambridge, MA, USA. E-mail: [email protected]Accepted 2008 January 9. Received 2007 October 18; in original form 2007 April 9 SUMMARY Rocks often develop fabric when subject to deformation, and this fabric causes anisotropy of physical properties such as viscosity and seismic velocities. We employ 2-D analytical solutions and numerical flow models to investigate the effect of anisotropic viscosity (AV) on the development of Rayleigh–Taylor instabilities, a process strongly connected to litho- spheric instabilities. Our results demonstrate a dramatic effect of AV on the development of instabilities—their timing, location, and, most notably, their wavelength are strongly affected by the initial fabric. Specifically, we find a significant increase in the wavelength of instability in the presence of AV which favours horizontal shear. We also find that an interplay between regions with different initial fabric gives rise to striking irregularities in the downwellings. Our study shows that for investigations of lithospheric instabilities, and likely of other mantle processes, the approximation of isotropic viscosity may not be adequate, and that AV should be included. Key words: Creep and deformation; Seismic anisotropy; Dynamics of lithosphere and mantle; Rheology: crust and lithosphere; Rheology: mantle. 1 INTRODUCTION The response of anisotropic materials to stress depends on the ori- entation of the stress relative to the orientation of the anisotropy. Anisotropy of seismic wave speed in rocks has been studied vigor- ously in the last decades, both in experimental (e.g. Zhang & Karato 1995) and theoretical work (e.g. Kaminski & Ribe 2001). It has been shown that the deformation of rocks and minerals leads to develop- ment of crystallographic preferred orientation (CPO), which leads to seismic anisotropy (Karato et al. 1998). In addition, rotation of grains and inclusions, alignment of microcracks or melt lenses, and layering of different phases all lead to the development of shape pre- ferred orientation (SPO), an important source for seismic anisotropy (e.g. Crampin 1978; Holtzman et al. 2003; Maupin et al. 2005). The anisotropic viscosity (AV) of earth materials has received less attention, but its effects are dramatic. Using laboratory exper- iments, Durham & Goetze (1977) showed that the strain rate of creeping olivine with pre-existing fabric depends on the orientation of the sample and can vary by up to a factor of 50. This is because the orientation of the sample relative to the applied stress deter- mines which slip systems are activated. In the experiments of Bai & Kohlstedt (1992) on high-temperature creep of olivine and those of Wendt et al. (1998) on peridotites, the measured strain rate depended strongly on the relative orientation of the applied stress to the sample crystallographic axis. Honda (1986) calculated the long-wavelength constitutive relations for a transversely isotropic material, and con- cluded that these can be characterized by two viscosities—a nor- mal viscosity (η N ), associated with principal stresses normal to the easy-shear planes, and a shear viscosity (η S ), associated with shear- ing parallel to the easy-shear planes. More theoretical work (e.g. Weijermars 1992; Mandal et al. 2000; Treagus 2003) was done to assess the AV of composite materials, depending on the geometry and the relative strength of each component. These studies imply that regions of the earth that are not likely to become anisotropic by means of dislocation creep and LPO development may exhibit AV due to the deformation of composite materials, such as most natu- ral rocks, and two-phase materials, such as partially molten rocks. Recently, Pouilloux et al. (2007) discussed the anisotropic rheology of cubic materials and the consequences for geological materials. A few geodynamic studies have examined the effect of AV on mantle flow. Richter & Daly (1978) and Saito & Abe (1984) used analytical solution methods to investigate the development of Rayleigh–B´ enard instabilities in a viscously anisotropic medium with specified easy-shear geometry, and found a connection be- tween the anisotropy of the fluid and the length-scales of the convec- tion cells. In a very instructive study a few years later, Christensen (1987) showed that the inclusion of AV affects two important man- tle flows—postglacial rebound and thermal convection. For exam- ple, Christensen (1987) pointed out a spatial offset between mass anomalies and the resulting geoid signal in the presence of AV, which may help to reconcile the argued mismatch between observed uplift history near ice sheet margins and models of strong viscosity strati- fication in the mantle. AV also leads to channelling of flow into low viscosity region such as hot rising plumes. Nonetheless, Christensen concluded that the actual effect of AV in the earth’s mantle would be much smaller, as the fabric required for creating AV would be oblit- erated by the highly time-dependent flow. However, the abundant evidence for seismic anisotropy in the earth and its strong correla- tion with tectonic processes and features suggest that large parts of the mantle maintain fabric for long times. Pre-existing mechanical C 2008 The Authors 1 Journal compilation C 2008 RAS
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April 10, 2008 0:54 Geophysical Journal International gji˙3731
Geophys. J. Int. (2008) doi: 10.1111/j.1365-246X.2008.03731.x
GJI
Sei
smol
ogy
Rayleigh–Taylor instabilities with anisotropic lithospheric viscosity
Einat Lev and Bradford H. HagerDepartment of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, Cambridge, MA, USA. E-mail: [email protected]
Accepted 2008 January 9. Received 2007 October 18; in original form 2007 April 9
S U M M A R YRocks often develop fabric when subject to deformation, and this fabric causes anisotropyof physical properties such as viscosity and seismic velocities. We employ 2-D analyticalsolutions and numerical flow models to investigate the effect of anisotropic viscosity (AV)on the development of Rayleigh–Taylor instabilities, a process strongly connected to litho-spheric instabilities. Our results demonstrate a dramatic effect of AV on the development ofinstabilities—their timing, location, and, most notably, their wavelength are strongly affectedby the initial fabric. Specifically, we find a significant increase in the wavelength of instabilityin the presence of AV which favours horizontal shear. We also find that an interplay betweenregions with different initial fabric gives rise to striking irregularities in the downwellings.Our study shows that for investigations of lithospheric instabilities, and likely of other mantleprocesses, the approximation of isotropic viscosity may not be adequate, and that AV shouldbe included.
Key words: Creep and deformation; Seismic anisotropy; Dynamics of lithosphere andmantle; Rheology: crust and lithosphere; Rheology: mantle.
1 I N T RO D U C T I O N
The response of anisotropic materials to stress depends on the ori-
entation of the stress relative to the orientation of the anisotropy.
Anisotropy of seismic wave speed in rocks has been studied vigor-
ously in the last decades, both in experimental (e.g. Zhang & Karato
1995) and theoretical work (e.g. Kaminski & Ribe 2001). It has been
shown that the deformation of rocks and minerals leads to develop-
ment of crystallographic preferred orientation (CPO), which leads
to seismic anisotropy (Karato et al. 1998). In addition, rotation of
grains and inclusions, alignment of microcracks or melt lenses, and
layering of different phases all lead to the development of shape pre-
ferred orientation (SPO), an important source for seismic anisotropy
(e.g. Crampin 1978; Holtzman et al. 2003; Maupin et al. 2005).
The anisotropic viscosity (AV) of earth materials has received
less attention, but its effects are dramatic. Using laboratory exper-
iments, Durham & Goetze (1977) showed that the strain rate of
creeping olivine with pre-existing fabric depends on the orientation
of the sample and can vary by up to a factor of 50. This is because
the orientation of the sample relative to the applied stress deter-
mines which slip systems are activated. In the experiments of Bai &
Kohlstedt (1992) on high-temperature creep of olivine and those of
Wendt et al. (1998) on peridotites, the measured strain rate depended
strongly on the relative orientation of the applied stress to the sample
crystallographic axis. Honda (1986) calculated the long-wavelength
constitutive relations for a transversely isotropic material, and con-
cluded that these can be characterized by two viscosities—a nor-
mal viscosity (ηN ), associated with principal stresses normal to the
easy-shear planes, and a shear viscosity (ηS), associated with shear-
ing parallel to the easy-shear planes. More theoretical work (e.g.
Weijermars 1992; Mandal et al. 2000; Treagus 2003) was done to
assess the AV of composite materials, depending on the geometry
and the relative strength of each component. These studies imply
that regions of the earth that are not likely to become anisotropic by
means of dislocation creep and LPO development may exhibit AV
due to the deformation of composite materials, such as most natu-
ral rocks, and two-phase materials, such as partially molten rocks.
Recently, Pouilloux et al. (2007) discussed the anisotropic rheology
of cubic materials and the consequences for geological materials.
A few geodynamic studies have examined the effect of AV
on mantle flow. Richter & Daly (1978) and Saito & Abe (1984)
used analytical solution methods to investigate the development of
Rayleigh–Benard instabilities in a viscously anisotropic medium
with specified easy-shear geometry, and found a connection be-
tween the anisotropy of the fluid and the length-scales of the convec-
tion cells. In a very instructive study a few years later, Christensen
(1987) showed that the inclusion of AV affects two important man-
tle flows—postglacial rebound and thermal convection. For exam-
ple, Christensen (1987) pointed out a spatial offset between mass
anomalies and the resulting geoid signal in the presence of AV, which
may help to reconcile the argued mismatch between observed uplift
history near ice sheet margins and models of strong viscosity strati-
fication in the mantle. AV also leads to channelling of flow into low
viscosity region such as hot rising plumes. Nonetheless, Christensen
concluded that the actual effect of AV in the earth’s mantle would be
much smaller, as the fabric required for creating AV would be oblit-
erated by the highly time-dependent flow. However, the abundant
evidence for seismic anisotropy in the earth and its strong correla-
tion with tectonic processes and features suggest that large parts of
the mantle maintain fabric for long times. Pre-existing mechanical
diamonds) and isotropic (black line, circles). The results agree with
the predictions from the analytical solution presented above—the
fastest growth rate for the horizontal fabric is at a longer wave-
length than that for the dipping fabric or for an isotropic layer, and
the curve is indeed flatter at longer wavelengths. The minimum
growth rate for a dipping fabric is at almost the same wavelength as
that for an isotropic material, again in agreement with the analytical
predictions. Fig. 3 shows the material distribution in the different
model configurations after the fastest drips have sunk half of the box
depth, as well as the approximate location of the initial perturbed
interface (yellow curve). These snapshots demonstrate clearly that
the wavelength of the instabilities developing in the initially hor-
izontal models is greater than of those developing in the initially
dipping models. This emphasizes the advantage gained by using
numerical experiments—the analytical solution gives insight into
the behaviour of instabilities at small amplitudes, while the numer-
ical experiments are essential for predicting the behaviour as the
flow progresses and instabilities of finite-amplitude develop.
3 L AT E R A L LY VA RY I N G A N I S O T RO P Y
Intrigued by the dramatic results for a simple model of a homo-
geneous anisotropic dense layer described above, we proceed and
use numerical experiments to examine the effect of including lateral
variations in the initial anisotropic fabric of the dense layer.
3.1 Model setup
Fig. 4 depicts the model geometry and initial and boundary condi-
tions. The model domain is again a rectangular box with an aspect
ratio of 1:6.4. The location and amplitude of the interface between
the layers is the same as in Section 2. Following the findings of Sec-
tion 2, we perturb the interface with a wavelength long enough to
allow deformation at a wide range of wavelengths to develop freely.
The dense layer now contains two anisotropic regions in the centre,
each 1.6 wide, and two isotropic regions of the same high density
near the edges. The anisotropic regions differ only by their initial
fabric orientation—one (shown in red) initially has a horizontal easy
shear direction, and the other has an easy shear direction initially
dipping at 45◦ (shown in yellow). The viscosity of the buoyant layer
is equal to the normal viscosity of the anisotropic layer. The shear
viscosity of the anisotropic material is a factor of 10 less than its
normal viscosity. We shift the anisotropic regions laterally in dif-
ferent models in order to change the phase between the viscosity
structure and the density interface perturbation. We then examine
the development of drips for each configuration.
3.2 Results—a heterogeneous upper layer
In Fig. 5, we show the instabilities that develop in our models. The
different panels depict models with different configurations of the
initial fabric domains, shown in red and yellow, as well as the results
for an isotropic model for comparison (Fig. 5a). We also show the
trace of the original density interface between the dense lithosphere
Figure 3. Material distribution for models with horizontal (left-hand panels) and dipping (right-hand panels) initial fabric of the dense top layer and various
initial interface deflection wavelengths, taken after the fastest downwellings sink past half the box depth. Colour denotes the materials—blue is the isotropic
buoyant material and red is the anisotropic denser material. The yellow curves show the approximate location of the initial density interface, exaggerated
April 10, 2008 0:54 Geophysical Journal International gji˙3731
Instabilities of anisotropic lithosphere 5
No slip
No slip
Free
slip
Fre
e s
lip ρ1
1
0
0.5
ρ2ρ2 ρ2 ρ2
Figure 4. A schematic description of the model geometry and initial conditions. The colours denote the densities and rheologies: blue—isotropic, ρ = 1,
η iso = 1, red—anisotropic with horizontal fabric, ρ = 1, δ = 0.1, yellow—anisotropic with dipping fabric, ρ = 1, δ = 10, cyan—isotropic, ρ = 0, η iso = 1).
There is no slip on the top and bottom boundaries, and free slip is allowed along the side walls. The thickness of the top layer and the amplitude of the interface
perturbation were exaggerated for clarity.
Figure 5. Material distribution in models with different configurations of initial anisotropic fabric taken after the fastest downwelling sinks over half the box
depth. Panel A shows the results for an isotropic model. The black cosine curve at a depth of 0.15 marks the original interface between the dense and buoyant
layers. The vertical dashed black lines show the deepest points of the original density interface, where the dense layer was thickest. Red material starts with
a horizontal fabric; Yellow material starts with a fabric dipping at 45◦. Blue materials are isotropic. Interestingly both panels (b) and (g), which start with
distinctly different material arrangements, show large downwellings comprised of both anisotropic materials, while others do not.
and the underlying mantle (black horizontal curve) and the loca-
tion of the deepest points of the initial perturbation of the density
interface (dashed vertical lines).
Several first-order observations can be made in Fig. 5. First, there
is a striking difference between the instabilities that develop in
the two anisotropic domains. Most notably, the wavelengths of the
downwellings that develop in the domain with easy horizontal shear
are much longer than the wavelengths in the dipping-fabric domains
or in the isotropic model (Fig. 5a). In addition, the domain which
starts with easy horizontal shear (red) develops instabilities faster
than the domain which starts with easy shear direction dipping at
45◦ (yellow). Next, for several situations, the fastest-growing down-
welling does not coincide with the locations of maximum thickness
of the dense layer, but is offset horizontally by up to 0.5 of the box
depth (Figs 5b and g). Finally, almost all of the fastest-growing in-
stabilities occur near the edges of the domain of horizontal easy
shear (excluding the case where the thickest part of the dense layer
was exactly in the centre of the domain of initial horizontal aniso-
torpy), but the instabilities that develop in the dipping easy shear
domain develop in its interior. Evidently, the initial fabric and its
lateral variations influence the flow significantly.
4 D I S C U S S I O N
Our models are set up in a non-dimensional manner, for generality.
It is interesting, though, to rescale the results to lithospheric dimen-
sions. The dense layer (top 15 per cent of the box) corresponds to
the viscously mobile part of the lithosphere, which is approximately
its lowest 40 km. The viscosity of the lithosphere is temperature-
dependent, and is believed to decrease exponentially with depth,
with a reasonable decay length of about 10 km (Molnar et al. 1998).
If we take the viscosity at the base of the lithosphere to be 1019 Pa s
(Hager 1991), then the average viscosity for a 40 km thick layer,
calculated as 〈η〉 = exp( log η1+log η2
2), is 7.4 × 1019 Pa s. Using the
thickness of the lower lithosphere as the characteristic length scale,
we can estimate the spacing between the isotropic instabilities as
130 km, and the wavelength of the longest anisotropic instabilities
is close to 400 km. The lateral offset between the downwellings
and the locations of maximum lithospheric thickness scales to a
maximum of approximately 150 km. We rescale velocities based
on the viscosity and density contrast, using the ‘Stokes Velocity’
(VStokes = ρ∗g∗h2
η, where η is the effective viscosity of the dense
layer, ρ is the density contrast and h is the dense layer thickness).
We estimate the difference between the density of the lower litho-
sphere and the density of the underlying asthenosphere as 40 kg m−3
(Molnar et al. 1998). After substituting the above values we can now
calculate the scaling of time. We estimate that the time it takes for
the drips to sink to a depth of 160 km (the stage shown in Fig. 5)
is approximately 12 Myr. This duration is within the range of times
estimated by Houseman & Molnar (1997) for removal of the base of
an unstable thickened lithosphere. This time and distance of sinking
imply an average sinking velocity of 14 mm yr−1.
The models we present here are a preliminary attempt at this
problem, and thus have some shortcomings when compared with
the lithosphere. First, the fabric development rule we use is a simple
rotational evolution law, and does not take into account factors such
as temperature, strain rate, and recrystallization, all known to affect
the development of CPO in rocks. Second, the rheology we use is a
April 10, 2008 0:54 Geophysical Journal International gji˙3731
8 E. Lev and B. H. Hager
σzz = 2ηps εzz (A1b)
σxz = ηss εxz, (A1c)
where η ps is a viscosity corresponding to pure shear stresses, and ηss
corresponding to simple-shear. For a material with a horizontal easy-
shear direction (horizontal layering, for instance) ηss is equivalent
to ηS defined in Section 2.1, η ps ≡ ηN , and ηss < η ps . For an
anisotropic material with a dipping easy-shear direction, ηS ≡ η ps <
ηss ≡ ηN . For an isotropic material, η ps = ηss . This constitutive
relation can be derived from a matrix form similar to that in eq. (4):
σ i j = 2ηN ε i j − 2(ηN − ηS)i jkl ε kl , where is an alignment
tensor reflecting the orientation of the symmetry axis. Then, the
transformation from a horizontal symmetry anisotropy to a dipping
symmetry can be achieved by a rotation of the 4th-order tensor .
In our analytical solution, we employ the propagator matrix tech-
nique (e.g. Hager & O’Connell 1981) to calculate the growth rate of
Rayleigh–Taylor instabilities as a function of the wavelength of the
density perturbation between the two materials. We set z = 0 at the
interface between the layers, z = 1 at the top of the dense layer, and
the initial location of the density interface as w = w0 cos(kx). For
the horizontal and 45◦-dipping orientations we consider here, this
definition of the interface perturbation leads to v x , σ zz ∝ cos(kx),
and v z , σ xz ∝ sin(kx), where k is the wavenumber. For other ori-
entations there may be a phase shift with depth (Christensen 1987).
Thanks to the orthogonality of the trigonometric basis functions,
we can write a simplified set of equations for each wavenumber. We
define a vector u = [v, u, σ zz , σ xz], where v is the vertical velocity, uis the horizontal velocity, σ zz is the normal stress in the z direction,
σ xz is the shear stress, and x and z are the horizontal and vertical
coordinates. After some manipulation, this definition of u enables
us to express the equations of flow in each layer for every k as
Du = Au + b, (A2)
where D = ∂
∂z , and b is a forcing term. The matrix A is where the
AV is manifested.
The definition of the anisotropic constitutive relation above leads
to a matrix A of the form:
A =
⎡⎢⎢⎢⎣
0 −k 0 0
k 0 0 η−1ss
0 0 0 −k
0 4ηpsk2 k 0
⎤⎥⎥⎥⎦ . (A3)
When η ps = ηss (isotropic material), the expression in (A3) is equal
to the matrix A given by Hager & O’Connell (1981). Otherwise,
it reflects the AV of the material by including the two different
viscosities.
The solution to eq. (A2) is of the form
u(z) = eA(z−z0)u(z0) +∫ z
z0
eA(z−ξ )b(ξ )dξ. (A4)
We define the propagator matrix P(z, z0) = eA(z−z0), so that the
velocities and stresses can be expressed as
u(z) = P(z, z0)u(z0) +n∑
i=1
P(z, ξi )b(ξi )ξI , (A5)
where ξ i is the depth at the centre of a the ith layer and ξ i is the
layer thickness. The propagator matrix for an anisotropic material
will naturally be different than the propagator matrix for an isotropic
material, given the difference in the corresponding A matrices. The
boundary conditions for our problem are no-slip at the top boundary
(z = 1), which we take to be the base of the rigid part of the litho-
sphere, and vanishing of the velocities and stresses as z → −∞.
We can express the boundary conditions using the vector u defined
earlier:
u(z = 1) = [0, 0, σ t
zz, σ txz
], u(z = −∞) = [0, 0, 0, 0].
(A6)
In order to fulfill the boundary condition as z → −∞, ujust below the interface has to be of the form u(z = 0−) =[C 1/2k, C 2/2k, C 1, C 2], where C 1, C 2 are the σ zz and σ xz at the
interface. We add a normalized forcing term which here represents
the gravitational forcing in the z direction. Thus u across the in-
terface, at the bottom of the dense layer, becomes u(z = 0+) =[C 1/2k, C 2/2k, C 1 + 1, C 2]. We propagate this u(z = 0+) upwards
to the top interface using the anisotropic propagator matrix P ani:
u(z = 1) = P ani u(z = 0+). From the no-slip boundary condition at
the top, the first two components of the resulting vector are equal to
zero. We now have two equations and two unknowns—C1 and C2.
We solve for these two unknowns and use the result to calculate the
vertical velocity at the interface.
The change in the interface location with time is equal to the
vertical velocity at the interface—v(z = 0), where v is the vertical
velocity. A result of the derivation described above is that the vertical
velocity at the interface is proportional to the perturbation of the
interface, that is:
v(z = 0) ≡ ∂w
∂t∝ w. (A7)
Therefore, the change in the interface depth follows an exponential
growth rule:w(z, t) = etτ , which gives the dependence of the growth
rate τ on the model parameters:
τ = 1
K (ηps, ηss, k), (A8)
K is a complicated function of the viscosities and the wavenumber,
of the form:
ρg × [a sum of exponents o f powers of ηN , ηs , and k]. The
exact expression is too long to give here explicitly, but can be
obtained using the Matlab code in the Supplementary Material
(Appendix S1). The resulting relationship between 1/K (τ ) for a
range of wavenumbers and a set of viscosity ratios is demonstrated
in Fig. 1; Fig. S1 (Supplementary Material) shows a similar calcu-
lation for a range of viscosity ratios and k = 0.1.
A careful inspection of the anisotropic matrix Aani and the
anisotropic propagator matrix P ani reveals a very interesting
phenomenon—an oscillatory behaviour with depth for certain vis-
cosity ratios. Let us define δ, the viscosity ratio, as δ = ηssηps
. As
we noted earlier, for a material with a horizontal easy-shear direc-
tion ηss < η ps , and thus δ < 1, while for a material with a dipping
easy-shear direction ηss > η ps and δ > 1. The eigenvalues of the
matrix A are used in the expression for the propagator matrix and
control the behaviour of the velocities and stresses in the medium.
For an isotropic material, these eigenvalues are real and repeated,
and the propagator matrix includes additional terms depending lin-
early on the depth—P ∝ (1 ± kz)e±kz (Hager & O’Connell 1981).
The anisotropic A matrix has, on the other hand, four distinct eigen-
values, of the form:
λi = ±k
(2 − δ ± 2
√1 − δ
δ
) 12
. (A9)
All the eigenvalues for a material with horizontal fabric (δ < 1)
are real, leading to a propagator matrix (and thus velocities and